1. **State the problem:** Factor the quadratic expression $$27q^2 + 144q + 192$$ completely. 2. **Identify the greatest common factor (GCF):** Look for the largest number that divides all coefficients. 3. The coefficients are 27, 144, and 192. The GCF is 3. 4. Factor out the GCF: $$27q^2 + 144q + 192 = 3(9q^2 + 48q + 64)$$ 5. **Factor the quadratic inside the parentheses:** We want to factor $$9q^2 + 48q + 64$$. 6. Since the leading coefficient is not 1, use the method of factoring by grouping or the AC method. 7. Multiply $$a \times c = 9 \times 64 = 576$$. 8. Find two numbers that multiply to 576 and add to 48. These numbers are 24 and 24. 9. Rewrite the middle term using these numbers: $$9q^2 + 24q + 24q + 64$$ 10. Group terms: $$(9q^2 + 24q) + (24q + 64)$$ 11. Factor each group: $$3q(3q + 8) + 8(3q + 8)$$ 12. Factor out the common binomial: $$(3q + 8)(3q + 8) = (3q + 8)^2$$ 13. **Write the complete factorization:** $$3(3q + 8)^2$$ **Final answer:** $$\boxed{3(3q + 8)^2}$$